\[ \int \frac {e^{-x} \left (e^x \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2 e^{-x} \left (e^{16}+\log (4)\right )}{x}} \left (e^x x^2+e^{16} \left (2+4 x+2 x^2\right )+\left (2+4 x+2 x^2\right ) \log (4)\right )\right )}{2 x^2+4 x^3+2 x^4} \, dx \]
Optimal antiderivative \[ x -\frac {{\mathrm e}^{\frac {2 \left (2 \ln \left (2\right )+{\mathrm e}^{16}\right ) {\mathrm e}^{-x}}{x}}}{2+2 x} \]
command
Int[(E^x*(2*x^2 + 4*x^3 + 2*x^4) + E^((2*(E^16 + Log[4]))/(E^x*x))*(E^x*x^2 + E^16*(2 + 4*x + 2*x^2) + (2 + 4*x + 2*x^2)*Log[4]))/(E^x*(2*x^2 + 4*x^3 + 2*x^4)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ x-\frac {2^{\frac {4 e^{-x}}{x}-3} e^{\frac {2 e^{16-x}}{x}-x} \left (x^2 \log (16)+4 x \log (4)+\log (16)\right )}{\left (\frac {e^{-x}}{x^2}+\frac {e^{-x}}{x}\right ) x^2 (x+1)^2 \log (2)} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {e^{-x} \left (e^x \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2 e^{-x} \left (e^{16}+\log (4)\right )}{x}} \left (e^x x^2+e^{16} \left (2+4 x+2 x^2\right )+\left (2+4 x+2 x^2\right ) \log (4)\right )\right )}{2 x^2+4 x^3+2 x^4} \, dx \]________________________________________________________________________________________